psubclsubN - Mathbox for Norm Megill

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Theorem psubclsubN 39897

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
psubclsub.s	⊢ 𝑆 = (PSubSp‘𝐾)
psubclsub.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
psubclsubN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

**Proof of Theorem psubclsubN**
Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
2		psubclsub.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3	1, 2	psubcli2N 39896	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋)
4		eqid 2740	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	4, 1, 2	psubcliN 39895	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋))
6	5	simpld 494	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7		psubclsub.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 7, 1	polsubN 39864	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
9	6, 8	syldan 590	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
10	4, 7	psubssat 39711	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
11	9, 10	syldan 590	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
12	4, 7, 1	polsubN 39864	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
13	11, 12	syldan 590	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
14	3, 13	eqeltrrd 2845	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ‘cfv 6573 Atomscatm 39219 HLchlt 39306 PSubSpcpsubsp 39453 ⊥_𝑃cpolN 39859 PSubClcpscN 39891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-proset 18365 df-poset 18383 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p1 18496 df-lat 18502 df-clat 18569 df-oposet 39132 df-ol 39134 df-oml 39135 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-psubsp 39460 df-pmap 39461 df-polarityN 39860 df-psubclN 39892

This theorem is referenced by: pclfinclN 39907

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